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The fuzzy-C-means (FCM) algorithm is one of the most famous fuzzy clus-tering algorithms, but it gets stuck in local optima. In addition, this algo-rithm requires the number of clusters. Also, the density-based spatial of the application with noise (DBSCAN) algorithm, which is a density-based clus-tering algorithm, unlike the FCM algorithm, should not be pre-numbered. If the clusters are specific and depend on the number of clusters, then it can determine the number of clusters. Another...

Topics: Clustering, Fuzzy clustering, DBSCAN.

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In this study, a modified model of HIV with therapeutic and preventive controls is developed. Moreover, a simple evaluation of the optimal control problem is investigated. We construct the Hamiltonian function by way of integrating Pontryagin’s maximal principle to achieve the point-wise optimal solution. The effects obtained from the version analysis strengthen public health education to a conscious population, PrEP for early activation of HIV infection prevention, and early treatment with...

Topics: HIV, Optimal control problem, Basic reproduction number, Nu- merical simulation.

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The nonlinear space time dynamics have been discussed in terms of a hyper-bolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel poly-nomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously dif-ferentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the...

Topics: Sine-Gordon equation, Bessel polynomials, Wave equation, Or- thogonal Collocation.

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The present study addresses an exponentially fitted finite difference method to obtain the solution of singularly perturbed two-point boundary value problems (BVPs) having a boundary layer at one end (left or right) point on uniform mesh. A fitting factor is introduced in the derived scheme using the theory of singular perturbations. Thomas algorithm is employed to solve the resulting tri-diagonal system of equations. The convergence of the presented method is investigated. Several model...

Topics: Singular perturbation problem, Stability and convergence, Fi- nite difference method.

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The aim of this article is to present a new method based on Lucas poly-nomials and residual error function for a numerical solution of fractional Bagley–Torvik equations. Here, the approximate solution is expanded as a linear combination of Lucas polynomials, and by using the collocation method, the original problem is reduced to a system of linear equations. So, the approximate solution to the problem could be found by solving this system. Then, by using the residual error function and...

Topics: Fractional Bagley–Torvik equation, Caputo derivative, Lucas polynomials, Residual error function,...

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In this paper, a new orthogonal system of nonpolynomial basis functions is introduced and used to solve a class of time-fractional partial differential equations that have nonsmooth solutions. In fact, unlike polynomial bases, such basis functions have singularity and are constructed with a fractional variable change on Hahn polynomials. This feature leads to obtaining more accurate spectral approximations than polynomial bases. The introduced method is a spectral method that uses the...

Topics: Fractional-order shifted Hahn functions, Fractional-time partial differential equations, Spectral...

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In this work, we present a new mathematical model for the spread of hepatitis C disease in two populations: human population and medical equipment population. Then, we apply the Jacobi wavelets method com-bined with the decoupling and quasi-linearization technique to solve this set of nonlinear differential equations for numerical simulation.

Topics: Hepatitis C, Sterilization, Jacobi wavelets, Operational matrix of derivative, Simulation.

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A singularly perturbed time delay parabolic problem with two small pa-rameters is considered. The paper develops a finite difference scheme that is exponentially fitted on a uniform mesh in the spatial direction and uses the implicit-Euler method to discretize the time derivative in the temporal direction in order to obtain a better numerical approximation to the solu-tions of this class of problems. We establish the parameter-uniform error estimate and discuss the stability of the suggested...

Topics: Singular perturbation, Two parameters parabolic convection- diffusion problem, Time delay, Fitted...

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In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equations using Caputo sense. We obtain the integral form of the problem and transform it into a system of linear alge-braic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method’s...

Topics: Integro-differential equations, Collocation method, Fredholm- Volterra equations, Multi- order.

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In this paper, a high order accuracy method is developed for finding the approximate solution of two-point boundary value problems. The present approach is based on a special algorithm, taken from Pascal’s triangle, for obtaining a generalized form of the parametric splines of degree (2k + 1), k = 1, 2, . . . , which has a lower computational cost and gives the better ap-proximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm. The approximate solution...

Topics: Boundary value problems, Parametric spline, Band matrices, Pascal’s triangle.

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The proposed study is focused to introduce a novel integral transform op-erator, called Generalized Bivariate (GB) transform. The proposed trans-form includes the features of the recently introduced Shehu transform, ARA transform, and Formable transform. It expands the repertoire of existing Laplace-type bivariate transforms. The primary focus of the present work is to elaborate fashionable properties and convolution theorems for the proposed transform operator. The existence, inversion, and...

Topics: Integral transform, Lane–Emden type differential equations, Wave-like partial differential...

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The fuzzy operations on fuzzy numbers of type L-R are much easier than general fuzzy numbers. It would be interesting to approximate a fuzzy number by a fuzzy number of type L-R. In this paper, we state and prove two significant application inequalities in the monotonic functions set. These inequalities show that under a condition, the nearest fuzzy number of type L-R to an arbitrary fuzzy number exists and is unique. After that, the nearest fuzzy number of type L-R can be obtained by solving a...

Topics: Fuzzy number of type L-R, Nearest trapezoidal fuzzy number, Shape function, Fuzzy approximation.

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In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the...

Topics: Fractional integral-differential equations, Numerical algorithms, Weakly singular kernels, Cubic...

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In this research, we aim to analyze a mathematical model of Maize streak virus disease as a problem of fractional optimal control. For dynamical analysis, the boundedness and uniqueness of solutions have been investi-gated and proven. Also, the basic reproduction number is obtained, and local stability conditions are given for the equilibrium points of the model. Then, an optimal control strategy is proposed for the purpose of examining the best strategy to fight the maize streak disease. We...

Topics: Fractional differential equation, Maize streak virus, Fractional- order optimal control, Sweep...

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This study addresses the inverse issue of identifying the space-dependent heat source of the heat equation, which is stated using the optimal con-trol framework. For the numerical solution of this class of problems, an approach based on shifted Legendre polynomials and the associated oper-ational matrix is presented. The approach turns the primary problem into the solution of a system of nonlinear algebraic equations. To do this, the temperature and heat source variables are enlarged in terms...

Topics: Inverse problems, Optimal control problem, Shifted Legendre polynomials (SLPs), Heat source,...

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We deal with some effective numerical methods for solving a class of nonlinear singular two-point boundary value Fredholm integro-differential equations. Using an appropriate interpolation and a q-order quadrature rule of integration, the original problem will be approximated by the non-linear finite difference equations and so reduced to a nonlinear algebraic system that can be simply implemented. The convergence properties of the proposed method are discussed, and it is proved that its...

Topics: Nonlinear Fredholm integro-differential equations, Singular two- point boundary value, Numerical...

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This study is aimed at performing a comprehensive numerical evalua-tion of the iterative solution techniques without memory for solving non-linear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes are be evaluated, in which the ability of the methods to solve different types of nonlinear equations is be studied. First, 26 different iterative...

Topics: Nonlinear scalar equations, Iterative method, Efficiency index, Order of convergence, Initial...

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The purpose of this paper is to design a fully discrete hybridized discon-tinuous Galerkin (HDG) method for solving a system of two-dimensional (2D) coupled Burgers equations over a specified spatial domain. The semi-discrete HDG method is designed for a nonlinear variational formulation on the spatial domain. By exploiting broken Sobolev approximation spaces in the HDG scheme, numerical fluxes are defined properly. It is shown that the proposed method is stable under specific mild conditions...

Topics: Coupled Burgers equations, hybridized discontinuous Galerkin method, stability analysis.

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In this article, we find a priori and a posteriori error estimates of the fixed point for the Picard iteration associated with a noncyclic contraction map, which is defined on a uniformly convex Banach space with a modulus of convexity of power type. As a result, we obtain priori and posteriori error estimates of Zlatanov for approximating the best proximity points of cyclic contraction maps on this type of space.

Topics: Fixed point, Noncyclic contraction map, Uniformly convex Ba- nach space, Modulus of convexity,...

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In this study, we explore the theoretical features of a multiobjective interval-valued programming problem with vanishing constraints. In view of this, we have defined a multiobjective interval-valued programming prob-lem with vanishing constraints in which the objective functions are consid-ered to be interval-valued functions, and we define an LU-efficient solution by employing partial ordering relations. Under the assumption of general-ized convexity, we investigate the optimality conditions...

Topics: Multiobjective interval-valued optimization problem, vanishing constraints, (weakly) LU-efficient...

It is important to note that mixed systems of first and second-kind Volterra–Fredholm integral equations are ill-posed problems, so that solving discretized system of such problems has a lot of difficulties. We will apply the regularization method to convert this mixed system (ill-posed problem) to system of the second kind Volterra–Fredholm integral equations (well-posed problem). A numerical method based on Chebyshev wavelets is suggested for solving the obtained well-posed...

Topics: Mixed systems of rst and second-kind Volterra{Fredholm in- tegral equations, Regularization method,...

The class of strong stochastic Runge–Kutta (SRK) methods for stochas tic differential equations with a commutative noise proposed by R¨ oßler (2010) is considered. Motivated by Komori and Burrage (2013), we design a class of explicit stochastic orthogonal Runge–Kutta Chebyshev (SROCKC2) meth ods of strong order one for the approximation of the solution of Itˆo SDEs with an m-dimensional commutative noise.The mean-square and asymptotic stability analysis of the newly proposed methods...

Topics: Stochastic differential equations, Runge{Kutta methods, Stochas- tic mean square stability, Stiff...

Let T = (V, E) be a tree with | V |= n. A 2-(k, l)-core of T is two subtrees with at most k leaves and with a diameter of at most l, which the sum of the distances from all vertices to these subtrees is minimized. In this paper, we first investigate the problem of finding 2-(k, l)-core on an unweighted tree and show that there exists a solution that none of (k, l)-cores is a vertex. Also in the case that the sum of the weights of vertices is negative, we show that one of (k, l)-cores is a...

Topics: Core, Facility location, Median subtree, Semi-obnoxious

In this paper, the 1-median location problem on an undirected network with discrete random demand weights and traveling times is investigated. The objective function is to maximize the probability that the expected sum of weighted distances from the existing nodes to the selected median does not exceed a prespecified given threshold. An analytical algorithm is proposed to get the optimal solution in small-sized networks. Then, by using the centrallimit theorem, the problem is studied in...

Topics: Facility location, 1-median problem probabilistic weights, probabilistic traveling times.

The aim of the current paper is to construct the shifted fractional-order Jacobi functions (SFJFs) based on the Jacobi polynomials to numerically solve the fractional-order pantograph differential equations. To achieve this purpose, first the operational matrices of integration, product, and pantograph, related to the fractional-order basis, are derived (operational matrix of integration is derived in Riemann–Liouville fractional sense). Then, these matrices are utilized to reduce the main...

Topics: Fractional pantograph differential equation, Fractional-order Jacobi functions, Operational...

In this paper, an explicit exact finite-difference scheme for the Huxley equation is presented based on the nonstandard finite-difference (NSFD) scheme. Afterwards, an NSFD scheme is proposed for the numerical solution of the Huxley equation. The positivity and boundedness of the scheme is discussed. It is shown through analysis that the proposed scheme is consistent, stable, and convergence. The numerical results obtained by the NSFD scheme is compared with the exact solution and some...

Topics: The Huxley equation, Nonstandard nite-difference scheme, Positivity and boundedness, Consistency,...

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is...

Topics: Approximation, Optimal shape design, Linear programming, Measure theory.

We introduce Ciric-generalized quasicontractive fuzzy mappings and pro vide the necessary and sufficient conditions of having a unique endpoint for such mappings. Then we introduce β-ψ-quasicontractive fuzzy mappings, es tablishing an endpoint result for them. Finally, we provide some results as an application.

Topics: Fuzzy endpoint, Ciri c-generalized, Quasicontractive fuzzy map- pings, Fuzzy approximate endpoint...

Many real-world problems occur under uncertainty. In this paper, we consider interval linear programming (ILP) which can be used to tackle un certainties. Several methods have been proposed by researchers, such as the best and worst cases, Two-step method (TSM), improved TSM, ILP, improved ILP, three-step method, and robust two-step method. First, we define feasibility and optimality conditions in ILP models and review some solving methods shortly, and then show that some solutions of the TSM...

Topics: Interval linear programming, TSM, Uncertainty, Aerosol.

We exploit the relationship between multiobjective integer linear problem (MOILP) and data envelopment analysis (DEA) to develop an approach to a resource reallocation problem. The general purpose of the mathematical formulation of this multicriteria allocation model based on DEA is to enable decision-makers to take into account the efficiency of units under control to allocate additional resources for a new period of operation. We develop a formal approach based on DEA and MOILP to find the...

Topics: Multicriteria decision aiding, Data envelopment analysis, Mul- tiobjective integer linear...

We apply the Adomian decomposition method (ADM) to obtain a subop timal control for linear time-varying systems with multiple state and control delays and with quadratic cost functional. In fact, the nonlinear two-point boundary value problem, derived from Pontryagin’s maximum principle, is solved by ADM. For the first time, we present here a convergence proof for ADM. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the...

Topics: Multiple time-delay systems, Pontryagin's maximum principle, Adomian decomposition method.

Variational models are one of the most efficient techniques for image denoising problems. A variational method refers to the technique of optimizing a functional in order to restore appropriate solutions from observed data that best fit the original image. This paper proposes to revisit the discrete total generalized variation (TGV ) image denoising problem by redefining the operations via the inclusion of a diagonal term to reduce the staircasing effect, which is the patchy artifacts usually...

Topics: Image denoising, Total variation, Staircasing effect, Total gen- eralized variation, Peak signal to...

We present a method to minimize locally Lipschitz functions. At first, a local quadratic model is developed to approximate a locally Lipschitz function. This model is constructed by using the ϵ-subdifferential. We minimize this local model and compute a search direction. It is shown that this direction is descent. We generalize the Wolfe conditions for finding an adequate step length along this direction. Next, the method is equipped with a quasi Newton approach to update the local model and...

Topics: Quasi-Newton method, Quadratic model, Line search algo- rithm, Locally Lipschitz functions.

We design a fast technique for fitting cubic B´ezier curves to the boundary of 2D shapes. The technique is implemented by means of the Nelder–Mead simplex procedure to optimize the control points. The natural attributes of the B´ezier curve are utilized to discover the initial vertex points of the Nelder–Mead procedure. The proposed technique is faster than traditional methods and helps to obtain a better fit with a desirable precision. The comparative analysis of our results describes...

Topics: Interpolation, Splines, Curve ftting, Nelder-Mead simplex method, Computer aided design, Computer...

In this study, an indirect method is proposed based on the Chebyshev pseudo-spectral method for solving optimal control problems governed by Burgers’ equation. Pseudo-spectral methods are one of the most accurate methods for solving nonlinear continuous-time problems, specially optimal control problems. By using optimality conditions, the original optimal control problem is first reduced to a system of partial differential equations with boundary conditions. Control and state functions are...

Topics: Burgers' equation, Optimal control, Chebyshev-Gauss-Lobatto nodes.

An efficient direct and numerical method has been proposed to approximate a solution of time-delay fractional optimal control problems. First, a class of discrete orthogonal polynomials, called Hahn polynomials, has been introduced and their properties are investigated. These properties are employed to derive a general formulation of their operational matrix of fractional integration, in the Riemann–Liouville sense. Then, the fractional derivative of the state function in the dynamic...

Topics: Delay fractional optimal control problems, Riemann{Liouville integration, Hahn polynomials,...

The aim of the present work is to introduce a method based on the Chebyshev polynomials for numerical solution of a system of Cauchy type singular integral equations of the first kind on a finite segment. Moreover, an estimation error is computed for the approximate solution. Numerical resultsdemonstrate the effectiveness of the proposed method.

Topics: System of singular linear integral equations, Orthogonal poly- nomials, Fourier series, Best...

We expand a new generalization of the two-dimensional differential trans form method. The new generalization is based on the two-dimensional differential transform method, fractional power series expansions, and conformable fractional derivative. We use the new method for solving a nonlinear con formable fractional partial differential equation and a system of conformable fractional partial differential equation. Finally, numerical examples are presented to illustrate the preciseness and...

Topics: Conformable fractional derivative, Differential transform method, two-dimensional differential...

We propose a new preconditioned global conjugate gradient (PGL-CG) method for the solution of matrix equation AXB = C, where A and B are sparse Stieltjes matrices. The preconditioner is based on the support graph preconditioners. By using Vaidya’s maximum spanning tree precon ditioner and BFS algorithm, we present a new algorithm for computing the approximate inverse preconditioners for matrices A and B and constructing a preconditioner for the matrix equation AXB = C. This preconditioner...

Topics: Krylov subspace methods, matrix equation, approximate in- verse preconditioner, global conjugate...

We propose a numerical scheme to solve a general class of time-fractional order telegraph equation in multidimensions using collocation points nodes and approximating the solution using double shifted Jacobi polynomials. The main characteristic behind this approach is to investigate a time-space collocation approximation for temporal and spatial discretizations. The applica bility and accuracy of the present technique have been examined by the given numerical examples in this paper. By means...

Topics: Time-fractional order telegraph equation, Shifted Jacobi polynomials, Gauss-Jacobi nodes, Matrix...

We investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional diffusion (FD) equation, where the interconnections are of Neumann type. We exploit the PDE back stepping method as a powerful tool for designing a controller to show the Mittag–Leffler stability of the FD-FODE cascade. Finally, numerical simulations are presented to verify the results.

Topics: Backstepping, Stability, Fractional-order cascaded systems.

Mathematical ecology and mathematical epidemiology are major fields in both biology and applied mathematics. In the present paper, a fourdimensional eco-epidemiological model with infection in both prey and preda tor populations is studied. It consists of susceptible prey, infected prey, susceptible predator, and infected predator. The functional response is assumed to be of Lotka–Volterra type. The behavior of the system such as the existence, boundedness, and stability for solutions and...

Topics: Predator-prey, optimal control, Stability, Infected model.

A general overview of the scheduling’s literature of some researches shows that among various factors, the priority rules and also the structure of projects are two main factors that can be affected on the performance of multidirectional scheduling schemes. In addition, a variation on the number of directors in scheduling schemes (e.g., single directional, bi-directional, and tri-directional scheduling scheme) produces different makespans. However, the question of when to move from the single...

Topics: Scheduling Priority rule, Heuristic algorithms, Multidirectional scheduling schemes, Resource...

Many phenomena in various fields of physics are simulated by parabolic partial differential equations with the nonlocal initial conditions, while there are few numerical methods for solving these problems. In this paper, the Ritz–Galerkin method with a new approach is proposed to give the exact and approximate product solution of a parabolic equation with the nonstandard initial conditions. For this purpose, at first, we introduce a function called satisfier function, which satisfies all the...

Topics: Nonlocal time weighting initial condition, Ritz–Galerkin method, Satisfier function, Bernstein...

We discuss the controllability and observability of time-invariant (continuous time) linear systems with interval coefficients using the notion of being full rank of interval matrices. The most important advantage of the proposed attitude is to consider these two essential concepts, that is, control lability and observability, in interval time-invariant linear systems, which, in turn, may play important roles in the analysis of uncertain systems. Some different definitions on to be full rank of...

Topics: controllability, Observability, Time-invariant linear systems with interval coefficients.

We propose a new approach for solving nonlinear Klein–Gordon and sine-Gordon equations based on radial basis function-pseudospectralmethod (RBF-PS). The proposed numerical method is based on quasiinterpolation of radial basis function differentiation matrices for thediscretization of spatial derivatives combined with Runge–Kutta time stepping method in order to deal with the temporal part of the problem.The method does not require any linearization technique; in addition, a new technique is...

Topics: Meshless method, Pseudospectral method, Radial basis functions, Klein-Gordon equation, sine-Gordon...

We suggest a convenient method based on the Fibonacci polynomials and the collocation points for solving approximately the Abel’s integral equation of second kind. Initially, the solution is supposed in the form of the Fibonacci polynomials truncated series with the unknown coefficients. Then, by placing this series into the main problem and collocating the resulting equation at some points, a system of algebraic equations is obtained. After solving it, the unknown coefficients and so the...

Topics: Abel’s integral equation, Fibonacci polynomials, Collocation points, Error analysis.

We utilize the homotopy analysis method to find eigenvalues of fractional Sturm–Liouville problems. Inasmuch as very few papers have been devoted to estimating eigenvalues of these kind of problems, this work enjoys a particular significance in many different branches of science. The convergence of the homotopy analysis method is also considered on the high order fractional Sturm–Liouville problem. The numerical results acknowledge the ability of the proposed method. Eigenvalues are...

Topics: Homotopy analysis method, Eigenvalues, Fractional Sturm– Liouville problems.

We present a novel algorithm, which is called Cutting Algorithm (CA), for improving the accuracy and reducing the computations of the Least Squares Support Vector Machines (LS-SVMs). The method is based on dividing the original problem to some subproblems. Since a master problem is converted to some small problems, so this algorithm has fewer computations. Although, in some cases that the typical LS-SVM cannot classify the dataset linearly, applying the CA the datasets can be classified. In...

Topics: Least squares support vector machine, Cutting algorithm, Classification.

Detecting the Pareto optimal solutions on the Pareto frontier is one of the most important topics in multiobjective optimal control problems. In real-world control systems, there is needed for the decision-maker to apply their own opinion to find the preferred solution from a large list of Pareto optimal solutions. This paper presents a class of axial preferred solutions for multiobjective optimal control problems in contexts in which partial information on preference weights of objectives is...

Topics: Multi-objective Optimal Control, Improvment Axis, Partial...

We consider the maximum flow network interdiction problem. We provide a new interpretation of the problem and define a concept called ”optimalcut”. We propose a heuristic algorithm to obtain an approximated cut, and we also obtain its error bound. Finally, we show that our heuristic is an α-approximation algorithm for a class of networks. By implementing it on three network types, we show the advantage of it over solving the model by CPLEX.

Topics: Interdiction, Approximation algorithm, Network flow, Minimum capacity cut.

Functionally graded materials (FGMs) are materials that show different properties in different areas due to the gradual change of chemical composition, distribution, and orientation, or the size of the reinforcing phase in one or more dimensions. In this paper, the free vibrations of a thin cylindrical shell made of FGM is investigated. In order to investigate this problem, the first-order shear theory is used, by using relations related to the propagation of waves and fluid-structure...

Topics: Functionally graded materials, Natural frequencies of cylindri- cal shell, First-order shear...

We apply a new method to solve fractional partial differential equations (FPDEs) with proportional delays. The method is based on expanding the unknown solution of FPDEs with proportional delays by the basis of Bernstein polynomials with unknown control points and uses operational matrices with the least-squares method to convert the FPDEs with proportional de lays to an algebraic system in terms of Bernstein coefficients (control points) approximating the solution of FPDEs. We use the Caputo...

Topics: Fractional partial differential equation, Bernstein polynomial, Operational matrix, Caputo...

We present a new numerical approach to solve the optimal control problems (OCPs) with a quadratic performance index. Our method is based on the Bell polynomials basis. The properties of Bell polynomials are explained. We also introduce the operational matrix of derivative for Bell polynomials. The chief feature of this matrix is reducing the OCPs to an optimization problem. Finally, we discuss the convergence of the new technique and present some illustrative examples to show the effectiveness...

Topics: Optimal control problems, Bell polynomial, Best approxima- tion, Operational matrix of derivative.

We consider a fully-discrete approximation of the Allen-Cahn equation, such that the forward Euler/Crank–Nicolson scheme (in time) combined with the RBF collocation method based on “shifted” surface spline (in space). Numerical solvability and stability of the method, by using second order finite difference matrices are discussed. We show that, in the proposed scheme, the nonlinear term can be treated explicitly and the resultant numerical scheme is linear and easy to implement. Numerical...

Topics: Allen{Cahn equation, RBF collocation method, Shifted surface spline, Stability, Solvability.

The most important purpose in location problems is usually to locate some facilities and allocate the demands of nodes so that the total transportation cost of the network is minimized. However, in real networks, there are some other influencing factors, aside from the transportation costs, for determin ing the allocation mode. In this paper, a minimum information approach is applied to the capacitated p-median problem to estimate the most likely allo cation solution based on some prior...

Topics: Network, Location problems, Capacitated p-median, Benders decomposition, Minimum information.

We introduce a new family of multivalue and multistage methods based on Hermite–Birkhoff interpolation for solving nonlinear Volterra integro differential equations. The proposed methods that have high order and ex tensive stability region, use the approximated values of the first derivative of the solution in the m collocation points and the approximated values of the solution as well as its first derivative in the r previous steps. Convergence order of the new methods is determined and...

Topics: Volterra integro-differential equations, Multistep collocation methods, Hermite–Birkhoff...

We propose a maximum probability model to estimate the origin-destination trip matrix in the networks, where the observed traffic counts of links and the target origin-destination trip demands are independent discrete random variables with known probabilities. The problem is formulated by using the least squares approach in which the objective is to maximize the probability that the sum of squared errors between the estimated values and the observed (target) ones does not exceed a pre-specified...

Topics: Transportation, Origin-destination trip matrix, Least squares approach, Probabilistic traffic...

In this paper, based on a discrete total variation model, a modified discretization of total variation (TV) is introduced for image processing problems. Two optimization problems corresponding to compressed sensing magnetic resonance imaging (MRI) data reconstruction problem and image denoising are proposed. In the proposed method, instead of applying isotropic TV whose gradient field is a two directions vector, a four directions discretization with some modification is applied for the inverse...

Topics: Total variation, Magnetic resonance imaging, Primal-dual opti- mization method, Regularization,...

We study two numerical techniques based on the homotopy perturba tion transform method (HPTM) and the fractional Adams–Bashforth method (FABM) for solving a class of nonlinear time-fractional differential equations involving the Caputo–Prabhakar fractional derivatives. In this manuscript, the convergence for numerical solutions obtained using HPTM and the con vergence and stability for numerical solutions obtained using FABM are inves tigated. We compare the solutions obtained by the HPTM...

Topics: Nonlinear time-fractional differential equations, Fractional Ho- motopy perturbation transform...

A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous at tempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system...

Topics: Forward-backward heat equation, Nonoverlapping domain de- composition, Finite difference,...

We apply a primal-dual simplex algorithm for solving the biobjective min imum cost-time network flow problem such that the total shipping cost and the total shipping fixed time are considered as the first and second objective functions, respectively. To convert the proposed model into a single-objective parametric one, the weighted sum scalarization technique is commonly used. This problem is a mixed-integer programming, which the decision variables are directly dependent together. Generally,...

Topics: Biobjective network flow, Minimum cost-time, Primal-dual al- gorithm, Fixed time.

We present a numerical method for solving linear and nonlinear fractional partial differential equations (FPDEs) with variable coefficients. The main aim of the proposed method is to introduce an orthogonal basis of twodimensional fractional Muntz–Legendre polynomials. By using these polynomials, we approximate the unknown functions. Furthermore, an operational matrix of fractional derivative in the Caputo sense is provided for computing the fractional derivatives. The proposed approximation...

Topics: Two-dimensional fractional Muntz–Legendre polynomials (2D- FMLPs), Fractional partial...

In numerical analysis, the process of fitting a function via given data is called interpolation. Interpolation has many applications in engineering and science. There are several formal kinds of interpolation, including linear interpolation, polynomial interpolation, piecewise constant interpo lation, trigonometric interpolation, and so on. In this article, by using Sigmoid functions, a new type of interpolation formula is presented. To il lustrate the efficiency of the proposed new...

Topics: Sigmoid function, Interpolation, Numerical integration, Quadrature formula, Numerical solution of...

Recently, a one-parameter extension of the Polak–Rebière–Polyak method has been suggested, having acceptable theoretical features and promising numerical behavior. Here, based on an eigenvalue analysis on the method with the aim of avoiding a search direction in the direction of the maximum magnification by a symmetric version of the search direction matrix, an adaptive formula for computing parameter of the method is proposed. Un der standard assumptions, the given formula ensures the...

Topics: Unconstrained optimization, Conjugate gradient method, Maximum magnification, Line search.

As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral differential equation. Since the integral part might bring some difficulties, we put forward a fairly new nu merical scheme based on the alternating...

Topics: Interest rate, Option pricing, Jump-diffusion models, Alternating direction implicit, Convergence.

The stability and Hopf bifurcation of a nonlinear mathematical model are described by the delay differential equation proposed by Wodarz for inter action between uninfected tumor cells and infected tumor cells with the virus. By choosing τ as a bifurcation parameter, we show that the Hopf bifurcation can occur for a critical value τ . Using the normal form theory and the center manifold theory, formulas are given to determine the sta bility and the direction of bifurcation and other...

Topics: Hopf bifurcation, Delay model, Oncolytic viruses, Tumor cells.

Using the approximate endpoint property, we describe a technique for exist ing solutions of the fractional q -differential inclusion with boundary value conditions on multifunctions. For this, we use an approximate endpoint result on multifunctions. Also, we give an example to elaborate on our results and to present the obtained results by fractional calculus.

Topics: Approximate endpoint, Fractional q-differential inclusion, Boundary value conditions.

We consider the eigenvalues of the fractional-order Sturm -- Liouville equation of the form \begin{equation*} -{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1], \end{equation*} with Dirichlet boundary conditions $$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$ where $q\in L^2(0,1)$ is a real-valued potential function. The method is used based on a Picard's iterative procedure. We...

Topics: Fractional Sturm–Liouville, Fractional calculus, Iterative methods, Eigenvalues

A fourth-order and rapid numerical algorithm, utilizing a procedure as Runge–Kutta methods, is derived for solving nonlinear equations. The method proposed in this article has the advantage that it, requiring no calculation of higher derivatives, is faster than the other methods with the same order of convergence. The numerical results obtained using the devel oped approach are compared to those obtained using some existing iterative methods, and they demonstrate the efficiency of the present...

Topics: Order of convergence, Newton–Raphson method, Householder iteration method, Nonlinear equations

The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations. The coefficient matrices of these linear systems have an structure, where is a symmetric positive definite (SPD) matrix and satisfies . We introduce an approximation for the SPD part , which is called , and then we show that the preconditioner has the Toeplitz-like structure and its...

Topics: Fractional diffusion equation, Toeplitz-like matrix, Krylov subspace methods, PGMRES.

In this paper, some monotonicity-preserving (MP) and positivity-preserving (PP) splitting methods for solving the balance laws of the reaction and diffusion source terms are investigated. To capture the solution with high accuracy and resolution, the original equation with reaction source term is separated through the splitting method into two sub-problems including the homogeneous conservation law and a simple ordinary differential equa tion (ODE). The resulting splitting methods preserve...

Topics: Balance laws, Splitting method, Monotonicity-preserving.

We use the Müntz Legendre wavelets and operational matrix to solve a system of fractional integro-differential equations. In this method, the system of integro-differential equations shifts into the systems of the algebraic equation, which can be solved easily. Finally, some examples confirming the applicability, accuracy, and efficiency of the proposed method are given.

Topics: System of fractional integro-differential equations, Caputo fractional derivative, Müntz Legendre...

We find a solution of an unknown time-dependent diffusivity a ( t ) in a linear inverse parabolic problem by a modified genetic algorithm. At first, it is shown that under certain conditions of data, there exists at least one solution for unknown a ( t ) in ( a ( t ) , T ( x, t )), which is a solution to the corresponding problem. Then, an optimal estimation for unknown a ( t ) is found by applying the least-squares method and a modified genetic algo rithm. Results show that an excellent...

Topics: Inverse parabolic problem, Existence, Uniqueness, Green’s function, Fixed point, Contraction...

Fuzzy multiobjective linear bilevel programming (FMOLBP) problems are studied in this paper. The existing methods replace one or some deterministic model(s) instead of the problem and solve the model(s). Doing this work, we lose much information about the compromise decision, and it does not make sense for the uncertain conditions. To overcome the difficulties, Zadeh’s extension principle is applied to solve the FMOLBP problems. Two crisp multiobjective linear three-level programming problems...

Topics: Fuzzy mathematical programming, Fuzzy number, Fuzzy multi objective bilevel programming, Extension...

In this part of the study, several benchmark problems are solved to evalu ate the performance of the existing strain-based membrane elements, which were reviewed in the first part. This numerical evaluation provides a basis for comparison between these elements. Detailed discussions are offered after each benchmark problem. Based on the attained results, it is con cluded that inclusion of drilling degrees of freedom and also utilization of higher-order assumed strain field result in higher...

Topics: Strain-based formulation, Higher-order strain field, Equilibrium condition, Numerical evaluation,...

Since the introduction of the finite element approach, as a numerical solution scheme for structural and solid mechanics applications, various for mulation methodologies have been proposed. These ways offer different advantages and shortcomings. Among these techniques, the standard displacement-based approach has attracted more interest due to its straightforward scheme and generality. Investigators have proved that the other strategies, such as the force-based, hybrid, assumed stress, and as...

Topics: Strain-based formulation, Higher-order strain field, Equilibrium condition, Numerical evaluation,...

This paper studies the linear optimization problem subject to a system of bipolar fuzzy relation equations with the max-product composition operator. Its feasible domain is briefly characterized by its lower and upper bound, and its consistency is considered. Also, some sufficient conditions are proposed to reduce the size of the search domain of the optimal solution to the problem. Under these conditions, some equations can be deleted to compute the minimum objective value. Some sufficient...

Topics: Bipolar Fuzzy Relation Equation, Linear Optimization, Max- Product Composition, Modified...

The utilization factor (UF) measures the ratio of the total resources’ amount required to the availability of resources’ amount during the life cycle of a project. In 1982, in the journal of Management Science, Kurtulus and Davis claimed that “If two resource-constrained problems for each type of resource have the same UF’s value in each period of time, then each problem is subjected to the same amount of delay provided that the same sequencing rule is used (If different tie-breaking...

Topics: Scheduling scheme, Priority rule, Multi-project environment, Resource measure.

A numerical method for solving Fredholm and Volterra integral equations of the second kind is presented. The method is based on the use of the Newton–Cotes quadrature rule and Lagrange interpolation polynomials. By the proposed method, the main problem is reduced to solve some nonlinear algebraic equations that can be solved by Newton’s method. Also, we prove some statements about the convergence of the method. It is shown that the approximated solution is uniformly convergent to the...

Topics: Fredholm integral equation, Volterra integral equation, Newton– Cotes quadrature rule, Lagrange...

We present two numerical approximations with non-uniform meshes to the Caputo–Fabrizio derivative of order α (0 < α < 1) . First, the L1 formula is obtained by using the linear interpolation approximation for constructing the second-order approximation. Next, the quadratic interpolation approximation is used for improving the accuracy in the temporal direction. Besides, we discretize the spatial derivative using the compact finite difference scheme. The accuracy of the...

Topics: Numerical approximations, Caputo–Fabrizio fractional deriva- tive, Diffusion equation, Advection...

Singularly perturbed robin type boundary value problems with discontinuous source terms applicable in geophysical fluid are considered. Due to the discontinuity, interior layers appear in the solution. To fit the interior and boundary layers, a fitted nonstandard numerical method is constructed. To treat the robin boundary condition, we use a finite difference formula. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme,...

Topics: Singularly perturbed problem, Robin type boundary value prob- lems, Discontinuous source term,...

We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrödinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it...

Topics: Quantum Equations, Optimal Control Problems, Legendre Wavelets Methods.

As an important duality result in linear optimization, the Goldman–Tucker theorem establishes strict complementarity between a pair of primal and dual linear programs. Our study extends this result into the framework of linear fractional optimization. Associated with a linear fractional program, a dual program can be defined as the dual of the equivalent linear program obtained from applying the Charnes–Cooper transformation to the given program. Based on this definition, we propose new...

Topics: Linear fractional optimization, Charnes–Cooper transforma- tion, Duality, Strict complementarity.

Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems, in which the differential equation arising from this method is solved by the Pontryagin’s maximum principle. In this paper, a set of hybrid methods based on explicit 6th-order RungeKutta method is presented for the FBSM solution of optimal control problems. Order of truncation error, stability region, and numerical results of the new hybrid methods were compared with those of the...

Topics: OCP, Stability analysis, Hybrid methods.

In this article, singularly perturbed differential difference equations having delay and advance in the reaction terms are considered. The highest-order derivative term of the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval (0 , 1] . For the small value of ε , the solution of the equation exhibits a boundary layer on the left or right side of the domain depending on the sign of the convective term. The terms with the shifts are approximated by...

Topics: Differential difference, Exponentially fitted, Singularly per- turbed problem, Tension spline,...

Image processing by partial differential equations (PDEs) has been an active topic in the area of image denoising, which is an important task in computer vision. In PDE-based methods for unprocessed image process ing, the original image is considered as the initial value for the PDE and the solution of the equation is the outcome of the model. Despite the advan tages of using PDEs in image processing, designing and modeling different equations for various types of applications have always been...

Topics: Partial Differential Equations, Image Processing, Image Denois- ing, Optimal Control.

The focus of this article is on the study of discrete optimal control problems (DOCPs) governed by time-varying systems, including time-varying delays in control and state variables. DOCPs arise naturally in many multi-stage control and inventory problems where time enters discretely in a natural fashion. Here, the Euler--Lagrange formulation (which are two-point boundary values with time-varying multi-delays) is employed as an efficient technique to solve DOCPs with time-varying multi-delays....

Topics: Discrete-time optimal control problem with time-varying delay, Euler–Lagrange equations,...

A linearly implicit difference scheme for the space fractional coupled nonlinear Schrödinger equation is proposed. The resulting coeﬀicient matrix of the discretized linear system consists of the sum of a complex scaled identity and a symmetric positive definite, diagonal-plus-Toeplitz, matrix. An eﬀicient block Gauss–Seidel over-relaxation (BGSOR) method has been established to solve the discretized linear system. It is worth noting that the proposed method solves the linear equations...

Topics: The space fractional Schrödinger equations, Toeplitz matrix, Block Gauss-Seidel over-relaxation...

In this paper, we present an eﬀicient method to solve linear time-delay optimal control problems with a quadratic cost function. In this regard, first, by employing the Pontryagin maximum principle to time-delay systems, the original problem is converted into a sequence of two-point boundary value problems (TPBVPs) that have both advance and delay terms. Then, using the continuous Runge–Kutta (CRK) method, the resulting sequences are recursively solved by the shooting method to obtain an...

Topics: Pontryagin maximum principle, Time-delay two-point bound- ary value problems, Time-delay optimal...

In this paper, we introduce two new schemes based on the global Hessen-berg processes for computing approximate solutions to low-rank Sylvester tensor equations. We first construct bases for the matrix and extended matrix Krylov subspaces by applying the global and extended global Hes-senberg processes. Then the initial problem is projected into the matrix or extended matrix Krylov subspaces with small dimensions. The reduced Sylvester tensor equation obtained by the projection methods can be...

Topics: Low-rank Sylvester tensor equation, Global Hessenberg process, Extended Global Hessenberg process,...

The complexity of solving differential equations in real-world applications motivates researchers to extend numerical methods. Among different numerical and semi-analytical methods for solving initial and boundary value problems, the differential transform method (DTM) has received no-table attention. It has developed and experienced generalizations for implementing other types of mathematical problems such as optimal control, calculus of variations, and integral equations. This review aims to...

Topics: Boundary value problems, Initial value problems, Differential Transform Method, Semi-analytical...

In this paper, a two-dimensional time-fractional telegraph equation is considered with derivative in the sense of Caputo and 1 < β < 2. The aim of this work is to extend the Crank–Nicolson method for this time-fractional telegraph equation. The stability and convergence of the numerical method are investigated. Also, the accuracy and efficiency of the proposed method are demonstrated by numerical experiments.

Topics: Time-Fractional Telegraph Equation, Crank–Nicolson Method, Stability, Convergence.

This paper aims to apply and investigate the compact finite difference methods for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of compact finite difference. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the...

Topics: Fractional Riccati equation, Caputo fractional derivative, Com- pact finite difference methods.

We propose a two-phase algorithm for solving continuous rank-one quadratic knapsack problems (R1QKPs). In particular, we study the solution structure of the problem without the knapsack constraint. In fact, an algorithm is suggested in this case. We then use the solution structure to propose an algorithm that finds an interval containing the optimal value of the Lagrangian dual of R1QKP. In the second phase, we solve the Lagrangian dual problem using a traditional single-variable optimization...

Topics: Quadratic Knapsack Problem, Line-Sweep Algorithm

There is significant interest in studying security games for defense op-timization and reducing the effects of attacks on various security systems involving vital infrastructures, financial systems, security, and urban safe-guarding centers. Game theory can be used as a mathematical tool to maximize the eﬀiciency of limited security resources. In a game, players are smart, and it is natural for each player (defender or attacker) to try to deceive the opponent using various strategies in order...

Topics: Security game, Deceptive resource, Mixed-integer program- ming, Fuzzy theory, Z-number.

Let be an -by- matrix with index and . In this paper, the problem of stagnation of the DGMRES method for the singular linear system is considered. We show that DGMRES has partial stagnation of order at least if and only if belongs to the the joint numerical range of matrices where is a compression of to the range of Also, we characterize nonsingular part of a matrices such that DGMRES does not stagnate for all . Moreover, a sufficient condition for non-existence of real stagnation...

Topics: Stagnation, DGMRES method, Singular systems.

shifted Legendre orthonormal polynomials (SLOPs) are used to approximate the numerical solutions of fractional optimal control problems. To do so, first, the operational matrix of the Caputo fractional derivative, the SLOPs, and Lagrange multipliers are used to convert such problems into algebraic equations. Also, the method is proposed for solving multidimensional problems. We obtained the error bound of the operational matrix in fractional...

Topics: Shifted Legendre orthonormal polynomials (SLOPs), Fractional optimal control problem (FOCP), Caputo...

We estimate a function f with N independent observations by using Leg- endre wavelets operational matrices. The function f is approximated with the solution of a special minimization problem. We introduce an explicit expression for the penalty term by Legendre wavelets operational matri ces. Also, we obtain a new upper bound on the approximation error of a differentiable function f using the partial sums of the Legendre wavelets. The validity and ability of these operational matrices are shown...

Topics: Legendre wavelet, Operational matrix, Wavelet approximation, Regression function, Error analysis.

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourth- order accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented...

Topics: Singularly perturbed, Second-order problems, Nonpolynomial splines, Convergence